Problem: Which of the definite integrals is equivalent to the following limit? $ \lim_{n\to\infty} \sum_{i=1}^n \cos\left(\dfrac{\pi}2+\dfrac{\pi i}{2n}\right)\cdot\dfrac{\pi}{2n}$ Choose 1 answer: Choose 1 answer: (Choice A) A $ \int_0^{\pi /2} \cos x\,dx$ (Choice B) B $ \int_{\pi /2}^{\pi} \cos x\,dx$ (Choice C) C $ \int_{0}^{\pi} \cos x\,dx$ (Choice D) D $ \int_{\pi /2}^{3\pi /4} \cos x\,dx$
Solution: The value of a definite integral is the limit of its Riemann sums as the number of terms tends to infinity. The given summation $ \sum_{i=1}^n \cos\left(\dfrac{\pi}2+\dfrac{\pi i}{2n}\right)\cdot\dfrac{\pi}{2n}$ looks like a right Riemann sum with $n$ subintervals of equal width. If each subinterval has width $\Delta x$, what is the right Riemann sum for the following definite integral? $ \int_a^b f(x) \,dx$ The right Riemann sum for the definite integral is $ \sum_{i=1}^n f(a+i\Delta x)\cdot\Delta x$. What does the sum become when we express $\Delta x$ in terms of $a$, $b$, and $n$ ? Dividing the interval $[a,b]$ into $n$ subintervals of equal width yields a common width of $\Delta x=\dfrac{b-a}n\,$. This lets us express the right Riemann sum as $ \sum_{i=1}^n f\left(a+i\cdot\dfrac{b-a}n\right)\cdot\dfrac{b-a}n$. Let's rewrite the given summation as $ \sum_{i=1}^n \cos\left(\dfrac\pi2 + i\cdot\dfrac{\pi/2}n\right) \cdot\dfrac{\pi/2}n\,$. If the given summation is a right Riemann sum, what are $a$, $b$, and $f$ ? Equating the width $\Delta x$ of each subinterval in the two sums yields $\dfrac{b-a}n=\dfrac{\pi/2}n\,$. Thus, the interval $[a,b]$ has width $b-a=\pi/2$. In all the answer choices, $f(x)=\cos x$. Therefore, $a=\pi/2$ and $b=a+\pi/2=\pi$. These values produce the definite integral $ \int_{\pi/2}^\pi \cos x \,dx$. The correct answer is $ \lim_{n\to\infty} \sum_{i=1}^n \cos\left(\dfrac\pi2+\dfrac{\pi i}{2n}\right)\cdot\dfrac{\pi}{2n} = \int_{\pi /2}^{\pi} \cos x \,dx$.